train/precalculus/1013.json

{
    "problem": "Find\n\\[\\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 3 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 5 \\\\ 0 & 1 \\end{pmatrix} \\dotsm \\begin{pmatrix} 1 & 99 \\\\ 0 & 1 \\end{pmatrix}.\\]",
    "level": "Level 3",
    "type": "Precalculus",
    "solution": "More generally,\n\\[\\begin{pmatrix} 1 & a \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & b \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 & a + b \\\\ 0 & 1 \\end{pmatrix}.\\]Therefore,\n\\[\\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 3 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 5 \\\\ 0 & 1 \\end{pmatrix} \\dotsm \\begin{pmatrix} 1 & 99 \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 & 1 + 3 + 5 + \\dots + 99 \\\\ 0 & 1 \\end{pmatrix} = \\boxed{\\begin{pmatrix} 1 & 2500 \\\\ 0 & 1 \\end{pmatrix}}.\\]"
}